Problem of the Week

Updated at Mar 9, 2015 9:39 AM

To get more practice in calculus, we brought you this problem of the week:

How can we find the derivative of xtan⁡x\frac{\sqrt{x}}{\tan{x}}?

Check out the solution below!



ddxxtan⁡x\frac{d}{dx} \frac{\sqrt{x}}{\tan{x}}

1
Use Quotient Rule to find the derivative of xtan⁡x\frac{\sqrt{x}}{\tan{x}}. The quotient rule states that (fg)′=f′g−fg′g2(\frac{f}{g})'=\frac{f'g-fg'}{{g}^{2}}.
tan⁡x(ddxx)−x(ddxtan⁡x)tan⁡2x\frac{\tan{x}(\frac{d}{dx} \sqrt{x})-\sqrt{x}(\frac{d}{dx} \tan{x})}{\tan^{2}x}

2
Since x=x12\sqrt{x}={x}^{\frac{1}{2}}, using the Power Rule, ddxx12=12x−12\frac{d}{dx} {x}^{\frac{1}{2}}=\frac{1}{2}{x}^{-\frac{1}{2}}
tan⁡x2x−x(ddxtan⁡x)tan⁡2x\frac{\frac{\tan{x}}{2\sqrt{x}}-\sqrt{x}(\frac{d}{dx} \tan{x})}{\tan^{2}x}

3
Use Trigonometric Differentiation: the derivative of tan⁡x\tan{x} is sec⁡2x\sec^{2}x.
tan⁡x2x−xsec⁡2xtan⁡2x\frac{\frac{\tan{x}}{2\sqrt{x}}-\sqrt{x}\sec^{2}x}{\tan^{2}x}

Done
Cymath foriOSAndroid